In the photographic printing art in order to automatically print color negatives and produce color prints, color exposure determination algorithm equations compute red, green and blue exposures for each negative. The automatic calculation of these exposures is based upon the densities measured in the negative to be printed.
The equations used in the algorithms are based on prior experience gained by printing many typical consumer negatives. One of the preferred methods used to convert that experience into equation form is least squares regression analysis. Least squares regression analysis may be formulated using matrix notation in the following way.
The densities measured in each negative to be printed are combined to form quantities called predictors. The average density could be one such predictor, and the maximum and minimum densities may be other such predictors. The predictors useful in calculating the required exposure are determined by those skilled in the art. The use of this invention is in no way limited to any particular set of predictors. For convenience they may be represented by names such as x.sub.1, x.sub.2 and x.sub.3. The following description uses three predictors, but any number of predictors may be employed. The predictor values from a series of negatives can be arranged in a matrix form which includes a column of 1's to represent the balance coefficients (which are later adjusted) in the equation. Such a matrix for N frames would take the form: ##EQU1## where the first subscript represents the negative number in the sequence of N negatives and the second subscript stands for the predictor number as described above.
If the rows and columns of the matrix X are interchanged, then the matrix formed is the transpose of X, designated X'. The product of the matrices X'X is known as the sums of squares and cross products matrix.
The exposure values calculated by the color exposure determination algorithm equations are known to those skilled in regression analysis as the predicted response variables, any one of which may be designated as y. The exposure values which are determined by an operator and which result in the optimum prints from negatives are known as the aim responses, the corresponding one of which may be called y. The predicted exposures, and the aim exposures, y, from the same series of N negatives as described above may be collected into two column vectors which take the form: ##EQU2## where the subscript represents the negative number in the sequence of N negatives.
Regression theory teaches that the vector of predicted response variables is computed from the equation: EQU Y=X [X'X].sup.-1 X'Y (1 )
where
[X'X].sup.-1 is the inverse of the sums of squares and cross products matrix. PA1 b=[X'X].sup.-1 X'Y and PA1 Y=Xb PA1 b =is the column vector of color exposure determination algorithm equation coefficients. PA1 H is called the "hat" matrix because it puts the "hat" on Y or converts Y to Y. This is also called an orthogonal projection operator. PA1 h.sub.i is the diagonal element of the ith row from the hat matrix, PA1 x.sub.i is the ith row of predictor values from the X matrix. PA1 p is the number of degrees of freedom used in the regression (i.e., the number of columns in the X matrix). PA1 (a) scanning each negative to provide red, green and blue density information; PA1 (b) producing prints of such negatives employing such density information in a color exposure determining algorithm having at least one neutral density equations with coefficients and at least two chromaticity equations with coefficients; PA1 (c) computing a statistic from each negative's density data based on the diagonal element from the algorithm's hat matrix; PA1 (d) alerting the operator that a negative should be critically reprinted whenever its computed statistic lies within an adjustable selection range: PA1 (e) automatically identifying the prints made from the negatives whose computed statistic lies within the adjustable "selection" range. PA1 (f) selecting identified prints and providing precise density and color balance corrections; PA1 (g) reprinting negatives from the selected identified prints using the precise corrections; PA1 (h) automatically examining all reprinted negatives for suitability of use for recalculating the algorithm's coefficients and selecting those negatives which lie in one adjustable and one nonadjustable range for the hat statistic; PA1 (i) computing the X'X, the X'Y and Y'Y matrices for all the selected negatives; PA1 (j) adding an adjustable fraction of the new X'X the new X'Y and the new Y'Y matrices to the corresponding current matrices and recomputing the algorithm coefficients, b when the number of new frames eceeds an adjustable number; PA1 (k) installing and using, the new coefficients in the corresponding equations; and PA1 (l) using the newly formed X'X matrix to continue to compute the hat statistic and begin to accumulate new values for the new X'X, the new X'Y and the new Y'Y matrices.
The matrix products [X'X].sup.-1 X'Y give the least squares estimates of the algorithm coefficients. Thus:
where
Furthermore, equation (1) may be split up in a different manner to define the "hat" matrix: EQU H=X (X'Y).sup.-1 X'
where
The diagonal elements of the hat matrix are given by: EQU h.sub.i =x.sub.i [X'X].sup.-1 x.sub.i ' (2)
where
Both h.sub.i and x.sub.i are values associated with the ith negative out of the collection of N negatives. The average value of h.sub.i for a collection of N negatives is EQU h.sub.i =p / N
where
As is well known to those skilled in regression analysis the h.sub.i 's have important diagnostic properties. In particular, when the value of h.sub.i exceeds some multiple of p / N, such as 2p / N, then the negative is said to have high leverage (influence) in determining the coefficients, b.
All of the preceeding are well known to those skilled in regression analysis. This analysis is described as if there were a single set of predictors and a single response, but it should be understood that the same calculations are repeated for each equation used in the exposure determination algorithm. Thus, if there were three equations required to compute the red, green and blue exposures, then there would be three sets of coefficients, b, three hat matrices, three hat statistics and all the calculations would be done three times. If the negatives were split into two groups and four equations were required, then the calculations would be done four times. If more groups and thus more equations were required, then more sets of calculations would be done.
A percentage of negatives printed by any automatic algorithm will be incorrectly printed. This percentage will be increased by a mismatch between the characteristics of the negatives being printed, the film or paper contrast, the preferences of the photofinishing operator and the coefficients of the predictors used in the algorithm. The negatives can, for example, due to seasonal variations, contain a higher percentage of high contrast flash scenes or low contrast snow scenes than the number for which the algorithm coefficients are optimum. Alternatively, the operator may run the process at a higher or lower contrast than that for which the algorithm coefficients are optimum. In addition the operator may prefer, for example, a different compromise for printing underexposed negatives than would result using the coefficients of the color exposure determination algorithm equations. In any event, an automatic algorithm performs better if its coefficients are calculated so as to provide optimum performance for the printing situation.
The resources required to develop and optimize a color exposure determination algorithm are substantial. These include means to collect a large number of representative negatives (over 1000 for each film type), means to scan (densitometer) and store the density data from them, means to print them and store the printing information, means to judge the prints and make the necessary corrections from the first printing, means to reprint the negatives until the optimum print is made, means to calculate the predictor values from the density data, means to collect and assemble the predictor values along with the optimum printing conditions data and means to analyze this data using conventional statistical methods such as regression analysis. In principle each photofinishing installation could undertake their own algorithm development and optimization by assembling the capabilities described above. In practice this is not practical.